Non-linearity and non stationarity of the New Year abnormal wave

Applied Ocean Research 30 (2008) 2 1 5 - 2 2 0

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o c e . k H

Applied Ocean Research

R E S E A 1

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journal homepage: www.elsevier.com/locate/apor

Non-linearity and non-stationarity of the New Year abnormal wave

Z. Cherneva, C. Guedes Soares *

Centre for Marine Technology and Engineering (CENTEC), Technical University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

A R T I C L E I N F O A B S T R A C T Article histoiy: Received 7 June 2007 Received in revised f o r m 4 August 2008 Accepted 12 August 2008 Available online 24 September 2008 Keywords:

Abnormal waves Rogue waves Freak waves

Time-frequency analysis High order spectra

Generalized time-frequency spectrum, time-frequency bispectrum and time-frequency trispectrum are calculated from the in-situ data record to investigate the non-stationarity and non-linearity of the New Year abnormal wave. A Choi-Williams kernel is used to reduce the aliasing. It is shown that for a short series of 7 waves the second peak of the time-frequency spectrum exists probably due to Benjamin-Feir instability. There are five peaks of energy during the abnormal wave In the time-frequency spectrum calculated over a 4 min series. Frequencies of two pairs of peaks are related as the frequencies of five-wave resonant interactions. The time-frequency bispectrum calculated for a 4 min long record near the abnormal wave is used to describe how the bispectrum of wind waves is changing in time. The result is compared with the stationary bispectrum ofthe record containing the abnormal wave investigated in an eariier study using the complete time series of 20 min. It is shown that phase coupling takes place in the front side of the abnormal wave. The time-frequency trispectrum is calculated too. It is found that before the abnormal wave the time-frequency trispectrum shows very intensive non-linear interactions. This process is stronger before the abnormal wave than after it.

© 2008 Elsevier Ltd. All rights reserved.

1. I n t r o d u c t i o n

P o w e r s p e c t r u m analysis used as a s t a n d a r d p r o c e d u r e i n o c e a n o g r a p h y is a p o w e r f u l t o o l f o r s t u d y a n d p r e d i c t i o n o f t h e p r o p e r t i e s o f w i n d g e n e r a t e d ocean waves. The i n t e n s i v e d e v e l o p m e n t o f t h e science r e l a t e d t o w i n d w a v e s i n t h e t w e n t i e t h c e n t u r y is closely c o n n e c t e d w i t h t h e successful a p p l i c a t i o n o f p o w e r f r e q u e n c y s p e c t r a [ 3 6 ] , w h i c h has b e e n a c o m m o n p r a c t i c e i n coastal a n d m a r i n e e n g i n e e r i n g a n d i n s h i p a n d o f f s h o r e s t r u c t u r e s design. The p o w e r f r e q u e n c y s p e c t r u m a n d t h e s e c o n d - o r d e r m o m e n t (or c u m u l a n t ) o f a s t a t i o n a r y process are a F o u r i e r t r a n s f o r m pair. T h e y describe t h e l i n e a r s t r u c t u r e o f the process. The idea t h a t t h e w i n d w a v e s are a s u p e r p o s i t i o n o f an i n f i n i t e n u m b e r o f i n d e p e n d e n t w a v e h a r m o n i c s is a n i m p l e m e n t a t i o n o f t h e m a t h e m a t i c a l i d e a l i z a t i o n f o r l i n e a r i t y a n d s t a t i o n a r i t y . I n t h i s case, a z e r o - m e a n Gaussian process, c o m p l e t e l y c h a r a c t e r i z e d b y its f i r s t t w o m o m e n t s , p r o v i d e s a l l a v a i l a b l e i n f o r m a t i o n a b o u t t h e s t r u c t u r e o f w i n d w a v e s . The a s s u m p t i o n s o f l i n e a r i t y a n d G a u s s i a n i t y m a k e a l l phase i n f o r m a t i o n t o be l o s t i n t h e p o w e r s p e c t r u m d e s c r i p t i o n . Because o f t h a t i t is c o n s i d e r e d t h a t t h e s p e c t r u m is phase b l i n d a n d a l l phases are r a n d o m w i t h a u n i f o r m d i s t r i b u t i o n . H o w e v e r , l i k e m o s t o f t h e processes e n c o u n t e r e d i n e n g i n e e r i n g p r a c t i c e , w i n d w a v e s are n o t e x a c t l y Gaussian n o r s t a t i o n a r y [ 4 ] .

* Corresponding author. Tel.: +351 218417957.

E-mail address: g u e d e s s ® m a r . i s t . u t l . p t (C. Guedes Soares). 0141-1187/$ - see f r o n t matter © 2008 Elsevier U d . A l l rights reserved. doi:10.1016/j.apor.2008.08.003 The h i g h e r o r d e r s t r u c t u r e o f t h e z e r o - m e a n , s t a t i o n a r y b u t n o n - G a u s s i a n w i n d w a v e process is t h e o r e t i c a l l y d e s c r i b e d i n clas-sical w o r k s o f H a s s e l m a n n [ 2 4 - 2 6 ] a n d L o n g u e t - H i g g i n s [ 3 8 ] . The f i r s t p a p e r p u b l i s h e d o n the s u b j e c t o f e x p e r i m e n t a l i n v e s t i g a t i o n o f t h e n o n - l i n e a r n a t u r e o f h i g h e r o r d e r w i n d w a v e s p e c t r a is due to H a s s e l m a n n et al. [ 2 7 ] . Since t h e n t h e b i s p e c t r u m has b e e n ap-p l i e d t o ocean w a v e s a n d e v e n t o o c e a n o g r a ap-p h y i n g e n e r a l .

The b i s p e c t r a o f s h o a l i n g surface g r a v i t y w a v e s have b e e n s t u d i e d b y Elgar a n d Guza [ 1 2 , 1 3 ] , w h i l e t h e b i s p e c t r a o f i n t e r n a l w a v e s have b e e n a n a l y z e d b y M c C o m a s a n d Briscoe [ 4 2 ] . The b i s p e c t r a o f d e e p - w a t e r w a v e s w e r e a n a l y s e d b y R o d r i g u e z et a l . [ 4 8 ] t o d e t e c t t h e presence o f phase c o u p l i n g b e t w e e n w a v e c o m p o n e n t s . The b i s p e c t r a a n d phase r e l a t i o n s b e t w e e n w a v e c o m p o n e n t s have b e e n analysed b y Cherneva a n d Guedes Soares, f o r s h a l l o w w a t e r [ 5 ] a n d f o r s t o r m sea states w i t h a b n o r m a l w a v e s [ 6 ] .

In m a n y events t h e a s s u m p r i o n o f s t a t i o n a r i t y o f w i n d w a v e s p r o v i d e s o n l y p a r t i a l i n f o r m a t i o n because i t suppresses t h e d e t a i l s o f local processes o c c u r r i n g at s h o r t t i m e scales like a b n o r m a l w a v e or w a v e g r o u p s t h a t are spectacular m a n i f e s t a t i o n s o f a c o m p l e t e l y c h a n g i n g sea state. N o n s t a t i o n a r i t y o f r e a l processes has m o t i v a t e d t h e d e v e l o p -m e n t o f a n u -m b e r o f t i -m e - f r e q u e n c y r e p r e s e n t a t i o n s t h a t describe the f r e q u e n c y c o n t e n t o f t h e s i g n a l as a f u n c t i o n o f t i m e . The m a i n d e v e l o p m e n t s o f m e t h o d s f o r c a l c u l a t i o n o f h i g h e r o r d e r spectra have o c c u r r e d w i t h i n t h e s t a t i s t i c a l c o m m u n i t y [ 2 , 3 ] a n d i n t h e area o f d i g i t a l s i g n a l p r o c e s s i n g a p p l i e d t o d i f f e r e n t t y p e s o f t i m e series analysis [ 1 7 , 1 8 , 4 4 ] . I f i t is necessary t o m a k e an i n t e r p r e t a t i o n o f t h e t i m e - f r e q u e n c y e n e r g y d i s t r i b u t i o n o f a s i g n a l , t h e n a q u a d r a t i c t i m e - f r e q u e n c y

216 Z. Cherneva, C. Guedes Soares / AppUed Ocean Research 30 (2008)215-220

r e p r e s e n t a t i o n m u s t be used. T h e y are k n o w n as b i l i n e a r t i m e - f r e q u e n c y r e p r e s e n t a t i o n s s u c h as the s p e c t r o g r a m , W i g n e r - V i l l e , Page, Rihachek, and C h o i - W i l l i a m s d i s t r i b u t i o n s [8, 3 0 , 4 1 , 4 5 ] , w a v e l e t s [ 4 0 , 3 5 ] a n d e m p i r i c a l m o d e d e c o m p o s i -t i o n [ 3 1 , 5 1 ] . A l l o f -these -t r a n s f o r m s express -t h e e v o l u -t i o n o f -the s e c o n d - o r d e r s p e c t r a l c o n t e n t o f t h e s i g n a l u n d e r analysis w i t h re-spect to t i m e . Some o f these m e t h o d s have b e e n used i n w i n d w a v e i n v e s t i g a t i o n s [5,23,51,35,40] and s o m e w e r e e v e n u s e d t o s t u d y sea states i n w h i c h a b n o r m a l w a v e s have o c c u r r e d [ 6 , 3 3 , 3 7 , 5 2 ] .

H o w e v e r , b i l i n e a r r e p r e s e n t a t i o n s c a n n o t g i v e i n f o r m a t i o n a b o u t t h e t e m p o r a l e v o l u t i o n o f t h e h i g h e r t h a n s e c o n d - o r d e r s p e c t r u m o f t h e s i g n a l . A t h i r d o r d e r W i g n e r d i s t r i b u t i o n w a s i n -t r o d u c e d f o r -the f i r s -t -t i m e b y Gerr [ 1 6 ] . This m i x e d -t i m e - f r e q u e n c y - f r e q u e n c y r e p r e s e n t a t i o n k n o w n as a W i g n e r b i s p e c t r u m e x t e n d s t h e W i g n e r d i s t r i b u t i o n i n t h e same w a y t h a t t h e s t a t i o n a r y bispec-t r u m e x bispec-t e n d s bispec-t h e s bispec-t a bispec-t i o n a r y p o w e r s p e c bispec-t r u m . F u r bispec-t h e r m o r e , h i g h e r o r d e r m o m e n t spectra are d e f i n e d b y Fonolosa a n d N i k i a s [ 1 4 , 1 5 ] as g e n e r a l i z a t i o n s o f Cohen's class o f t i m e f r e q u e n c y r e p r e -s e n t a t i o n -s , w h o al-so -s h o w t h e d i f f e r e n c e -s b e t w e e n m o m e n t and c u m u l a n t t i m e - f r e q u e n c y h i g h e r o r d e r s p e c t r a . The h i g h e r o r d e r t i m e - f r e q u e n c y c u m u l a n t spectra w e r e i n t r o d u c e d b y G i a n n a k i s a n d D a n d a w a t e [ 1 9 ] .

The t i m e - f r e q u e n c y m e t h o d s t h a t are a v a i l a b l e a l l o w t h u s the s t u d y o f t h e t i m e e v o l u t i o n o f v a r i o u s p r o p e r t i e s o f d i f f e r e n t t y p e s o f processes, i n c l u d i n g ocean waves, a n d can t h u s be a n i m p o r t a n t t o o l t o s t u d y l o c a l i z e d p h e n o m e n a s u c h as w a v e g r o u p s or e v e n the n e i g h b o r h o o d o f i n d i v i d u a l w a v e s .

The a b n o r m a l , r o g u e or f r e a k w a v e s have b e e n o f m u c h c o n c e r n i n t h e last f e w years, w i t h v a r i o u s specialists t r y i n g t o u n d e r s t a n d t h e v a r i o u s aspects t h a t c a n be i n v o l v e d . T h i s t y p e o f w a v e has b e e n i d e n t i f i e d at d i f f e r e n t l o c a t i o n s as r e p o r t e d f o r e x a m p l e i n [ 2 0 , 2 1 , 2 9 , 4 3 ] a n d several m e c h a n i s m s have b e e n s u g g e s t e d as p o s s i b l e causes o f t h e i r o c c u r r e n c e as r e v i e w e d b y K h a r i f a n d P e l i n o v s k y , [ 3 2 ] .

The basic p r o b l e m starts w i t h t h e i d e n t i f i c a d o n o f w h a t is an a b n o r m a l w a v e . D r a p e r , w h o w a s p r o b a b l y t h e f i r s t to m e n t i o n t h i s t y p e o f w a v e c a l l e d t h e m f r e a k w a v e s [ 1 1 ] . T h e f i r s t o n e to p r o p o s e a c r i t e r i o n f o r its i d e n t i f i c a t i o n w a s p r o b a b l y D e a n [ 1 0 ] , w h o d e f i n e d a n a b n o r m a l w a v e i n l i g h t o f t h e l i n e a r t h e o r y , as a single w a v e o f h e i g h t e x c e e d i n g t w o t i m e s t h e s i g n i f i c a n t w a v e h e i g h t , n a m e l y H m a x / H i / 3 > 2. T h i s r a t i o is k n o w n as t h e a m p l i f i c a t i o n o r a b n o r m a l i t y i n d e x , A l . H a v e r [ 2 8 ] , o n t h e o t h e r h a n d , a s s u m e d second o r d e r t h e o r y f o r t h e u n d e r l y i n g p o p u l a t i o n o f w a v e s i n m o d e r a t e l y n o n l i n e a r sea w i t h s t a t i o n a r y a n d h o m o g e n e o u s c o n d i t i o n s a n d d e f i n e d the f r e a k w a v e as b e i n g t h e o u t l i e r i n t h a t p o p u l a t i o n .

Petrova et al. [ 4 6 , 4 7 ] s t u d i e d v a r i o u s w a v e records a n d o b s e r v e d t h a t the sea states w i t h i d e n t i f i e d a b n o r m a l w a v e s s h o w e d large d i s c r e p a n c y f r o m t h e l i n e a r a n d s e c o n d - o r d e r m o d e l s i n the e x t r e m e tails o f t h e d i s t r i b u r i o n s . T h e crests associated w i t h t h e a b n o r m a l w a v e s appear as o u t l i e r s i n t h e u n d e r l y i n g p o p u l a t i o n s o f n o n l i n e a r crest events.

So, evidence is s h o w i n g t h a t a t h e o r y h i g h e r t h a n second o r d e r m a y be necessary t o e x p l a i n t h e o c c u r r e n c e o f t h e a b n o r m a l w a v e s . I t is i n t h i s b a c k g r o u n d t h a t the p r e s e n t p a p e r a i m s at s t u d y i n g t h e t e m p o r a l e v o l u t i o n o f t h e l o w e r o r d e r n o n - l i n e a r spectra o f t h e a b n o r m a l w a v e s . For t h i s p u r p o s e t h e t i m e - f r e q u e n c y second-, t h i r d - a n d f o u r t h - o r d e r m o m e n t s p e c t r a are used t o i n v e s t i g a t e s h o r t - t i m e p r o p e r t i e s o f t h e s t o r m sea state i n w h i c h t h e N e w Year a b n o r m a l w a v e o c c u r r e d . 2. H i g h e r o r d e r t i m e - f r e q u e n c y s p e c t r u m Let X ( t ) r e p r e s e n t an a n a l y t i c a l s i g n a l . A p p l y i n g t h e f o r m u l a t i o n g i v e n i n [ 8 ] f o r t h e g e n e r a l class o f t i m e - f r e q u e n c y b i l i n e a r d i s t r i b u t i o n s i n [ 1 4 , 1 5 ] a g e n e r a l class o f t i m e - f r e q u e n c y h i g h e r o r d e r spectra is d e f i n e d as: ^HOSGEN ( t j u •••Jk) = I n ••• <PHOS (^2, T, , . . . , tfc) R,„ J J J ^2IIT^ J X ( T , , . . . , Tk) e x p (27vju^^). e x p { - I n j t Q ) k X ] ~ [ e x p ( - 2 ; r j 7 i T , ) d r , d i ( d ^ 2 ( 1 ) i = i w h e r e Rku ( t i , T2 tk) is a / ( d i m e n s i o n a l local m o m e n t f u n c -t i o n a n d 0HOS ( i 2, TI , . . . , T,() is a k e r n e l f u n c t i o n i n t r o d u c e d to r e -duce t h e a l i a s i n g i n t h e c a l c u l a t e d t i m e - f r e q u e n c y h i g h e r o r d e r s p e c t r a The p r o p e r t i e s o f t h e m o s t a p p l i e d b i l i n e a r t i m e - f r e q u e n c y d i s t r i b u t i o n s can be o b t a i n e d f r o m (1) f o r /( = 1 u s i n g d i f f e r e n t kernels cp (^2, T ) . I n p a r r i c u l a r , W i g n e r - V i l l e d i s t r i b u t i o n c o r r e s p o n d s t o (p (Q, r ) = 1, Rihachek t o 4> {Q, x) = e x p ijnxQ) a n d the C h o i - W i l l i a m s t o 0 ( ^ 2 , T ) = e x p ( - X J ^ r V o " ) w h e r e

a = 0 . 0 5 . Here t h e C h o i - W i l l i a m s k e r n e l is used i n its fc-th o r d e r

f o r m [ 7 , 1 4 ] : < ^ ( f i , T , , . . . , T k ) = e x p ( - ^ 2 2 ( r 2 + . . . + T k 2 ) / f f ) . ( 2 ) Thus i n t h i s w o r k t h e C h o i - W i l l i a m s d i s t r i b u t i o n e x t e n d e d t o t i m e - f r e q u e n c y h i g h e r o r d e r m o m e n t s p e c t r u m is a d o p t e d . For s i m p l i c i t y f u r t h e r i n t h e p a p e r " T F - s p e c t r u m " , " T F - b i s p e c t r u m " a n d " T F - t r i s p e c t r u m " w i l l be used, f o r / ( . e q u a l t o 1, 2 o r 3 i n Eq. ( 1 ) r e s p e c t i v e l y , to a v o i d t o o l o n g exact names o f t h e f u n c t i o n s c o n s i d e r e d . For p r a c t i c a l s i g n a l p r o c e s s i n g a p p l i c a t i o n o f the t i m e - f r e q u e n c y h i g h e r o r d e r s p e c t r a a d i s c r e t i z a t i o n i n t i m e a n d f r e q u e n c y is necessary. A d i s c r e t e v e r s i o n o f t h e t i m e f r e q u e n c y b i l i n e a r d i s t r i b u t i o n w a s i n i t i a l l y d e f i n e d i n [ 9 ] a n d l a t e r i n [ 4 5 ] . A d i s c r e t e v e r -s i o n o f t h e t i m e - f r e q u e n c y h i g h e r o r d e r -spectra i-s d e r i v e d i n [ 1 4 ] . The r e l a r i o n s b e t w e e n c o n t i n u o u s a n d d i s c r e t e v e r s i o n s o f t h e t i m e - f r e q u e n c y h i g h e r o r d e r spectra as w e l l as t h e i r n o n a l i a s i n g c o n s t r a i n t s are also g i v e n t h e r e . In t h e p r e s e n t p a p e r o n l y s y m m e t r i c d e f i n i t i o n s o f t h e t i m e - f r e q u e n c y h i g h e r o r d e r spectra have b e e n c o n s i d e r e d . I t w a s s h o w n i n [ 1 5 ] t h a t f o r t h e T F - t r i s p e c t r u m t h i s d e f i n i t i o n a l l o w s e f f i c i e n t c r o s s - t e r m a t t e n u a t i o n . The p r i n c i p a l d i a g o n a l s o f s t a t i o n a r y h i g h e r o r d e r s p e c t r a c o n t a i n t h e a u t o - t e r m i n f o r m a t i o n . O t h e r d i a g o n a l s c o n t a i n i n f o r m a t i o n a b o u t t h e c o u p l i n g b e t w e e n t h e d i f f e r e n t c o m p o n e n t s o f t h e s i g n a l . T h e t i m e - f r e q u e n c y h i g h e r o r d e r s p e c t r a are v e r y c o m p l i c a t e d a n d i t is d i f f i c u l t t o v i s u a l i z e t h e m . It is p o s s i b l e to p r e s e n t o n l y a t w o d i m e n s i o n a l slice o f t h e (/( + 1 ) -d i m e n s i o n a l t i m e - f r e q u e n c y h i g h e r o r -d e r s p e c t r u m . T h i s slice has to c o r r e s p o n d t o a p l a n e d e f i n e d t o the t e m p o r a l axis a n d one f r e q u e n c y l i n e p r e s e n t i n g j o i n t l y a l l / ( f r e q u e n c y axes. A m o n g a l l possible slices, t h e r e is one t h a t c o r r e s p o n d s t o t h e t e m p o r a l axis a n d t h e p r i n c i p a l d i a g o n a l o f t h e t i m e - f r e q u e n c y h i g h e r o r d e r s p e c t r u m . This p r i n c i p a l slices c o r r e s p o n d to / i = ƒ2 a n d ƒ, = ƒ2 = - ƒ 3 i n t h e t i m e - f r e q u e n c y planes o f the b i s p e c t r u m a n d the t r i s p e c t r u m r e s p e c t i v e l y . It is p r o v e d i n [ 1 5 ] t h a t o f f - p r i n c i p l e slices c o n t a i n o n l y c r o s s - t e r m s . 3. E x p e r i m e n t a l r e s u l t s

3.1. Full scale wave data

This w o r k has a n a l y s e d d a t a f r o m t h e j a c k e t p l a t f o r m " D r a u p -ner'' s i t u a t e d i n t h e C e n t r a l N o r t h Sea. T h e w a t e r d e p t h at t h a t l o c a t i o n is a b o u t 70 m a n d t h e w a v e s are m e a s u r e d b y a d o w n w a r d - l o o k i n g laser device d u r i n g 20 m i n w i t h N = 2 5 6 0 a n d t i m e s a m p l i n g rate A t = 0 . 4 6 8 7 5 7 3 s. I n pai t i c u l a r o n J a n u a r y 1

Z, Cherneva, C. Guedes Soares/AppUed Ocean Research 30 (2008) 215-220 217 C e n t r a l N o r t h S e a - 1 9 9 5 0 1 O i l 5 2 0

5 0 0 1 0 0 0

Fig. 1. Time series of sea surface elevation i n Draupner f r o m Jan. 1 1995 at 15:20.

TF-spectrum

0

0 20 40 60 80 100 120 140 160 180 200 Time [in samples]

Fig. 2. TF-spectrum of the "New Year Wave" for 7 waves near the giant wave.

1995 an a b n o r m a l h i g h w a v e w a s r e c o r d e d i n t h e t i m e series t h a t s t a r t e d at 15.20 h o f t h a t day (Fig. 1), as r e p o r t e d f o r the f i r s t t i m e b y H a v e r a n d K a r u n a k a r a n [ 2 9 ] .

This w a v e , w i t h an a b n o r m a l l y h i g h crest, w h i c h has b e c o m e k n o w n as t h e " N e w Year W a v e " , has been s t u d i e d b y d i f f e r e n t a u t h o r s t h a t are a i m i n g at a b e t t e r u n d e r s t a n d i n g a b o u t r o g u e o r a b n o r m a l w a v e s [ 4 9 , 5 2 ] .

A n analysis t h a t is closely r e l a t e d t o the p r e s e n t has been m a d e b y K u r k i n and Pelinovsl<y [ 3 4 ] . Some p a r a m e t e r s o f t h e " N e w Year W a v e " are e s t i m a t e d i n t h e above w o r k : kh ^ 2, ka ^ 0.37, a n d a/h ^ 0.2, w h e r e k is t h e w a v e n u m b e r v e c t o r , a is t h e a m p l i t u d e a n d h is t h e w a t e r d e p t h . K u r k i n , a n d P e l i n o v s k y [ 3 4 ] c o n c l u d e d t h a t t h e r e g i s t e r e d a b n o r m a l w a v e is s t r o n g l y n o n l i n e a r T h e y c o n s i d e r e d the i n f l u e n c e o f t h e B e n j a m i n a n d Feir i n s t a b i l i t y , e s t i m a t e d the m a x i m u m i n s t a b i l i t y w a v e n u m b e r ksp ~ 0 . 0 0 8 a n d p r o v e d t h a t 7 w a v e s are e n o u g h t o m a n i f e s t t h e m o d u l a t i o n i n s t a b i l i t y . This f a c t is d e m o n s t r a t e d b y the t i m e - f r e q u e n c y s p e c t r u m c a l c u l a t e d o v e r t h e w a v e series o f 83 s t h a t c o n t a i n s 7 w a v e s . The s p e c t r a l m a x i m u m is s p l i t j u s t b e f o r e a p p e a r i n g o f t h e a b n o r m a l w a v e . The s p l i t t i n g has been n o t o b s e r v e d f o r s h o r t e r series.

3.2. TF-spectrum

C a l c u l a t i n g t h e T F - s p e c t r u m o f t h e " N e w Year W a v e " o v e r t h e series o f 7 w a v e s o n e can observe s i m i l a r results as i n [ 3 4 ] . A s m a l l p a r t o f t h e w a v e r e c o r d consists o f a s h o r t g r o u p o f t h r e e w a v e s w i t h r e l a t i v e l y large periods a n d a f o u r - w a v e g r o u p w i t h the a b n o r m a l w a v e , as can be observed i n Fig. 2.

H e r e t h e s u r f a c e e l e v a t i o n is p r e s e n t e d as a w h i t e c o l o u r a n d f o r the sake o f c o n v e n i e n c e its t i m e axis coincides w i t h t h e s p e c t r a l one. The T F - s p e c t r u m is n o r m a l i s e d b y its h i g h e s t peak a n d t h e r e s u l t is p r e s e n t e d i n a p l a n e b y 10% isolines w h e r e t h e s u r f a c e b e t w e e n t h e m is f i l l e d w i t h d i f f e r e n t c o l o u r s . The m a x i m u m peak is at ^ 0.5 r a d / s , w h i c h c o n v e r t e d t o t i m e gives a p e r i o d o f

Tp = 12.57 s. This is e x a c t l y t h e p e r i o d o f t h e g i a n t w a v e . T h e

d i f f e r e n c e b e t w e e n t h e f r e q u e n c i e s o f t h e f i r s t a n d t h e s e c o n d p e a k is Aa> ^ 0.1 r a d / s a n d is i n t h e same range as t h e f r e q u e n c y i n t e r v a l g i v e n i n [ 3 4 ] .

F u r t h e r m o r e , t h e T F - s p e c t r u m is c a l c u l a t e d u s i n g a 4 m i n series t h a t c o n t a i n s 2 0 w a v e s . The results are p r e s e n t e d i n Fig. 3. There are f i v e peaks o f e n e r g y d u r i n g the o c c u r r e n c e o f t h e a b n o r m a l w a v e . T h e i r f r e q u e n c i e s are w , = 0 . 3 6 0 r a d / s , 0)2 = 0 . 4 3 2 r a d / s , 0)3 = 0 . 4 8 6 r a d / s , 0)4 = 0 . 5 4 0 r a d / s , a n d cos = 0.648 r a d / s . The peaks o n t h e s a m e f r e q u e n c i e s e x i s t i n t h e T F - s p e c t r u m b u i l t u s i n g o t h e r parts o f t h e same r e c o r d b u t these peaks are l o w e r t h a n t h e o n e o c c u r r i n g d u r i n g t h e large w a v e . Some o f t h e f r e q u e n c i e s a b o v e are c o n n e c t e d b y s i m p l e r e l a t i o n s : w i = 3 / 2 w 4 a n d C02 = 3/2dt>5.

218 Z. Cherneva, C. Guedes Soares / Applied Ocean Research 30 (2008) 215-220 Stationary Bispectrum Principal Slice

1.2f - • - • T F - B i s p e c t r u m " 0.6 Ll- 0.4 0.8 0.6 0.4 0.2 Normalised Bispectrum 100 200 300 400

Time [in samples]

Fig. 4. Principal slices of stationary bispectrum and Wigner bispectrum.

In a w e a k l y n o n l i n e a r a p p r o a c h A n n e n k o v a n d B a d u l i n [ 1 ] use t h e so-called f i v e - w a v e r e d u c e d Z a k h a r o v e q u a t i o n to s h o w t h a t o n e o f t h e possible r e s o n a n t i n t e r a c t i o n s are o f t h e t y p e 3/2((jL>-^- u> + Ü) - a> - cü). This a l l o w s t h e c o n c l u s i o n t h a t p r o b a b l y t h e pairs , cu^ a n d « 2 , « 5 ai'e t h e r e s u l t o f f o u r t h - o r d e r n o n - l i n e a r r e s o n a n t i n t e r a c t i o n b e t w e e n the w a v e c o m p o n e n t s .

The p o w e r s p e c t r u m d r a w n o n t h e l e f t side o f Fig. 3 is c a l c u l a t e d f o r t h e w h o l e o f t h e r e c o r d w i t h l e n g t h o f 2 0 m i n . I t has t w o peaks at f r e q u e n c i e s a>\, a n d 0)4. The peak o n 0)3 is n o t w e l l d e f i n e d . There is n o peak at f r e q u e n c y 0)5 a n d p r o b a b l y the r e s o n a n t i n t e r a c t i o n

ü)2, COS takes place o n l y d u r i n g the occurrence o f t h e a b n o r m a l

w a v e . This suggests t h a t p a r t o f t h e energy o f t h e second peak o f C04 o f t h e s t a t i o n a r y s p e c t r u m is a r e s u l t o f f o u r t h o r d e r n o n -l i n e a r r e s o n a n t i n t e r a c t i o n s o f w a v e s w i t h f r e q u e n c i e s n e a r t h e f r e q u e n c y o f the s p e c t r u m m a x i m u m .

3.3. TF-bispectrum

It is k n o w n t h a t s e c o n d - o r d e r n o n l i n e a r i n t e r a c t i o n s are n o t r e s o n a n t . T h e y p r o d u c e the so-called f o r c e d w a v e s . The i n t e r a c t i o n s occur i f o n e o f the f r e q u e n c i e s is the s u m ( d i f f e r e n c e ) o f the o t h e r t w o : a>s = ± wi, a n d t h e same h o l d s f o r t h e phases: (p-i = tp\ ± ( f j . The r e l a t i o n s are k n o w n as a q u a d r a t i c phase c o u p l i n g . The phases (p-i a n d are i n d e p e n d e n t r a n d o m variables, u n i f o r m l y d i s t r i b u t e d i n the i n t e r v a l ( 0 , 2 ; T ) b u t ^03 is r e l a t e d w i t h tpi a n d cpj a n d is n o t i n d e p e n d e n t . Because o f t h a t , t h e phase d i s t r i b u t i o n d i f f e r s f r o m t h e u n i f o r m one as o b s e r v e d i n [5,6, 5 0 ] . O n l y p h a s e - c o u p l e d c o m p o n e n t s c o n t r i b u t e t o the t h i r d o r d e r c u m u l a n t s o f t h e process [ 4 4 ] . Thus, t h e s t a t i o n a r y b i s p e c t r u m appears t o be a u s e f u l t o o l f o r q u a d r a t i c phase c o u p l i n g d e t e c t i o n a n d d i s t i n c t i o n o f phase d e p e n d e n t c o m p o n e n t s f r o m those t h a t are n o t . O n the c o n t r a r y t h e p o w e r s p e c t r u m is phase b l i n d a n d does n o t possess s u c h a b i l i t i e s .

I n t h e l e f t side o f Fig. 4 t h e d i a g o n a l ( p r i n c i p a l ) slice o f t h e s t a t i o n a r y b i s p e c t r u m is p r e s e n t e d [ 6 ] . It has t w o peaks at f r e q u e n c i e s t h a t c o i n c i d e w i t h f r e q u e n c i e s wi = 0 . 3 6 0 r a d / s a n d

C04 = 0 . 5 4 0 r a d / s o f t h e s t a t i o n a r y p o w e r s p e c t r u m . The t h i r d

peak is at a> = 0.443 r a d / s a n d i t does n o t c o i n c i d e w i t h a n y peak o f t h e p o w e r s p e c t r u m . O n t h e r i g h t side o f t h e same figure is p r e s e n t e d a t i m e d e v e l o p m e n t o f the p r i n c i p a l slice o f t h e TF-b i s p e c t r u m e s t i m a t e d u s i n g t h e same p a r t o f t h e r e c o r d o f 4 m i n l e n g t h c o n t a i n i n g the a b n o r m a l w a v e l i k e i n Fig. 2. A d d i t i o n a l l y , t h e 2 0 m i n series is d i v i d e d i n t o several parts o f 4 m i n l e n g t h t h a t do n o t c o n t a i n the a b n o r m a l w a v e a n d the T F - b i s p e c t r u m is c a l c u l a t e d f o r t h e m t o o . The m o s t i n t e n s i v e q u a d r a t i c phase c o u p l i n g exists d u r i n g the " N e w Year W a v e " .

In t h i s process are i n v o l v e d a w i d e i n t e r v a l o f f r e q u e n c i e s . The h i g h e s t peaks are e x a c t l y at t h e s a m e f r e q u e n c i e s w i a n d W4 m e n t i o n e d above l i k e the m a i n peaks o f t h e s t a t i o n a r y b i s p e c t r u m d i a g o n a l slice. The peak at t h e f r e q u e n c y 0)4 exists o n l y d u r i n g t h e a b n o r m a l w a v e a n d is n o t n o t i c e d i n t h e T F - b i s p e c t r u m c a l c u l a t e d o v e r the parts w i t h o u t t h e a b n o r m a l w a v e . It can be c o n c l u d e d t h a t t h i s peak o f the s t a r i o n a r y b i s p e c t r u m is a re.suIt o f t h e a b n o r m a l w a v e .

A n o t h e r i n t e r e s t i n g f e a t u r e o f t h e T F - b i s p e c t r u m is t h a t t h e q u a d r a t i c phase c o u p l i n g takes place i n the f r o n t side o f t h e g i a n t w a v e . The same can be seen f o r t h e w a v e b e f o r e t h e a b n o r m a l w a v e a n d f o r t h e t w o w a v e s a f t e r i t . I n an e a r i i e r w o r k [ 2 2 ] i n v e s t i g a t i n g t h e largest waves i n series w i t h s i g n i f i c a n t w a v e h e i g h t larger t h a n 4 m i t w a s f o u n d t h a t u s u a l l y t h e largest w a v e s are a s y m m e t r i c a n d t i l t e d to t h e f r o n t . T h e y are steeper i n its f r o n t side t h a n i n its back side. This t y p e o f a s y m m e t r y has been discussed b e f o r e by L o n g u e t -H i g g i n s [ 3 9 ] . It is s u p p o s e d t h e r e t h a t a s h e l t e r i n g e f f e c t g i v i n g a pressure d i f f e r e n c e rise o v e r t h e u p w i n d and l e e w a r d side o f the crests caused the w a v e a s y m m e t r y . Here, w i t h o u t a n y i n f o r m a t i o n a b o u t t h e w i n d , i t is f o u n d t h a t i n t h e f r o n t side o f the w a v e s i n t h e v i c i n i t y o f the a b n o r m a l w a v e t h e q u a d r a t i c phase c o u p l i n g is s i g n i f i c a n t .

3.4. TF-trispectrum

F r o m the d e f i n i t i o n ( 1 ) o f t h e T F - t r i s p e c t r u m f o r /< = 3 i t can be seen t h a t i t is a f o u r t h - o r d e r m o m e n t s p e c t r u m . Fig. 5 presents a p r i n c i p a l slice o f the T F - t r i s p e c t r u m o f t h e " N e w Year W a v e " . The r e l a t i o n a),,_i = w, Op+'i is o n l y b e t w e e n f r e q u e n c i e s .

The m e a s u r e m e n t is m a d e i n one p o i n t a n d i t is i m p o s s i b l e to say a n y t h i n g a b o u t t h e d i r e c t i o n s o f the i n t e r a c t i n g w a v e c o m p o n e n t s . On the o t h e r h a n d i t is o f n o use to m a k e a n o t h e r slice, because o f f -p r i n c i -p a l slices o f t h e T F - t r i s -p e c t r u m c o n t a i n o n l y c r o s s - t e r m s a n d t h e a l i a s i n g f o r those slices is s i g n i f i c a n t This m e a n s t h a t b u i l d i n g t h e T F - t r i s p e c t r u m w i l l p r o d u c e i n f o r m a t i o n a b o u t a special k i n d o f t h i r d - o r d e r w a v e i n t e r a c t i o n a n d w i l l a l l o w k n o w l e d g e about w h e n these i n t e r a c t i o n s take place. This m a k e s u p f o r all t h e losses o f p r i n c i p a l slice usage.

It can be seen i n Fig. 5 t h a t n e a r l y 20 s b e f o r e the a b n o r m a l w a v e i n t e n s i v e t h i r d - o r d e r w a v e i n t e r a c t i o n s e x i s t . It is i n t e r e s t i n g to h i g h l i g h t t h a t n o n l i n e a r i n t e r a c t i o n s are f o r t h e same f o u r f r e q u e n c i e s t h a t are i n t h e t i m e - f r e q u e n c y s p e c t r u m b u t d u r i n g t h e a b n o r m a l w a v e . Here, d u r i n g t h e a b n o r m a l w a v e o c c u r r e n c e t h i r d - o r d e r i n t e r a c t i o n s are at f r e q u e n c i e s , wi = 0 . 3 6 0 r a d / s , a n d C04 = 0 . 5 4 0 r a d / s f o r w h i c h t h e r e l a t i o n oi, = 3 / 2 w 4 h o l d s . These i n t e r a c t i o n s are n o t so i n t e n s i v e as 2 0 s b e f o r e t h e a b n o r m a l w a v e .

Z. Cherneva, C Guedes Soares/ Applied Ocean Research 30 (2008) 215-220 ₂₁₉

r u w u , o p B ü i . u r i i Principal Sliced T i m e - f r e q u e n o y Trispectrum

''•2 " " - ' ' 1.2 . . . . , . , , j • j i j \

h

1

...-./kMlÊMkZ.-.'/.'

Fig. 5. Frequency spectrum and TF-trispectrum.

4 . C o n c l u s i o n s I n t h i s w o r k an a t t e m p t is m a d e f o r t h e f i r s t t i m e t o f i n d d i r e c t l y t h e m a i n f r e q u e n c i e s t h a t take place i n t h e f o r m a t i o n o f t h e a b n o r m a l " N e w Year W a v e " . T i m e - f r e q u e n c y k-th o r d e r m o m e n t s p e c t r a are u s e d f o r t h a t p u r p o s e . I t is f o u n d t h a t d u r i n g t h e a b n o i m a l w a v e s e v e r a l n o n l i n e a r m e c h a n i s m s are t a k i n g place. T h e B e n j a m i n a n d Feir i n s t a b i l i t y is m a n i f e s t e d f o r a s m a l l p a r t o f t h e r e c o r d c o n t a i n i n g seven w a v e s . T h e T F - s p e c t r u m d u r i n g t h e a b n o r m a l w a v e ( c a l c u l a t e d f o r 4 m i n s e r i e s ) c o n t a i n s five m a i n f r e q u e n c i e s . T w o pairs o f t h e m are five-wave r e s o n a n t i n t e r a c t i o n s r e l a t e d . Q u a d r a t i c phase c o u p l i n g is v e r y i n t e n s i v e i n t h e f r o n t s i d e o f t h e w a v e s f r o m t h e g r o u p o f t h e g i a n t w a v e . I n t e n s i v e t h i r d - o r d e r n o n l i n e a r i n t e r a c t i o n s o c c u r 2 0 s b e f o r e t h e a b n o r m a l w a v e . F r e q u e n c i e s o f these i n t e r a c t i o n s c o i n c i d e w i t h t h e h i g h e s t peak f r e q u e n c i e s o f t h e T F - s p e c t r u m . T h e s e c o n d p e a k o f t h e s t a t i o n a r y s p e c t r u m is p r o b a b l y a r e s u l t o f f o u r t h - o r d e r n o n l i n e a r i n t e r a c t i o n s o f s p e c t r a l c o m p o n e n t s n e a r t h e s p e c t r a l p e a k . A c k n o w l e d g e m e n t s T h e first a u t h o r has b e e n f u n d e d b y FCT, t h e P o r t u g u e s e F o u n -d a t i o n f o r Science a n -d T e c h n o l o g y , u n -d e r g r a n t S F R H / B P D / 3 0 5 8 9 / 2 0 0 6 . R e f e r e n c e s

[ l ] Annenkov S, Badulin S. M u l t i w a v e resonances and f o r m a t i o n o f h i g h -a m p l i t u d e w-aves i n the oce-an. In: Ol-agnon M , Ath-an-assoulis G, editors. Rouge waves 2000. Ifremer; 2 0 0 1 . p. 205-13.

|2] Brillinger D. A n i n t r o d u c t i o n to polyspectra. A n n Math Statist 1965;36: 1351-74.

[3] Brillinger D, Rosenblatt M . Computation and integration o f A:-th order spectra. In: Harris B. editor. Spectral analysis of t i m e spectra. NY: John Wiley&Sons-1967. p. 189-232.

[4] Cherneva Z. W i n d wave i n the coastal zone as a non-linear probability process. In: Ozhan E, editor. Proc o f t h e int. MEDCOAST conf. o n w i n d wave climate of the mediterranean and the black sea. Ankara: NATO Science for Stability Program Publ.; 1999. p. 7 1 - 8 0 .

[5] Cherneva Z, Guedes Soares C. Bispectra and t i m e - f r e q u e n c y spectra of w i n d waves i n the coastal zone. In: Guedes Soares C. Garbatov J, Fonseca N , editors. M a r i t i m e transportation and e x p l o i t a t i o n o f ocean and coastal resources, vol. 2. London: Taylor and Francis Group; 2005. p. 1005-14.

[6] Cherneva Z, Guedes Soares C Estimation o f the bispectra and phase d i s t r i b u t i o n o f storm sea states w i t h abnormal waves. Ocean Eng 2007;34: 2 0 0 9 - 2 0 .

[7] Choi H, W i l l i a m s W . Improved t i m e f r e q u e n c y representation o f m u l t i c o m p o -nent signals using expo-nential kernels. IEEE Trans Acoust Speech Signal Pro-cess 1989;37:862-71.

[8] Cohen L. Time frequency distnbutions. A review. Proc IEEE 1 9 8 9 ; 7 7 : 9 4 1 - 8 1 . [9] Claasen T, Meeklenbrauker W . The W i g n e r d i s t r i b u t i o n . A tool f o r

t i m e - f r e q u e n c y signal analysis; Part I I ; Discrete t i m e signals. Phillips 1 Res 1980;35:276-300.

110) Dean R. A b n o r m a l waves: A possible explanation. In: T o r u m A, Gudmestat 0, editors. Water wave kinematics. Kluwer; 1990. p. 6 0 9 - 1 2 .

[11] Draper L. Freak ocean waves. Oceanus 1964;10:13-5.

112] Elgar E, Guza R. Observations o f bispectra o f shoaling surface gravity waves. J Fluid Mech 1985;161:425-48.

113] Elgar E, Guza R. Nonlinear model predictions o f bispectra o f shoaling surface gravity waves. J Fluid Mech 1986; 167:1-18.

]14] Fonolosa J, Nikias C. W i g n e r higher order m o m e n t spectra: D e f i n i t i o n , properties, c o m p u t a d o n and application to transient signal analysis. IEEE Tran Signal Process 1993;41:245-66.

[15] Fonolosa J, Nikias C. Analysis o n finite-energy signals using higher order moments and spectra-based t i m e - f r e q u e n c y distnbutions. Signal Process 1994;36:315-28.

[16] Gerr N. Introducing a t h i r d - o r d e r W i g n e r d i s t r i b u t i o n . Proc IEEE 1988;76: 2 9 0 - 2 .

[17] Giannakis G, Mendel J. Identification o f n o n - m i n i m u m phase systems using higher order statistics. IEEE Trans Acoust Speech Signal Process 1989;37: 360-77.

[18] Giannakis GB, Mendel J. Cumulant-based order d e t e r m i n a t i o n of non-Gaussian ARMA models. IEEE Trans Acoust Speech Signal Process 1990;38:1411-23. [19] Giannakis GB, Dandawate AV. Polyspectral analysis o f nonstationary signals:

Bases, consistency and HOS-WV. In: Proc signal processing w o r k s h o p higher orderStat. 1991. p. 167-70.

]20] Guedes Soares C, Cherneva Z, Antao E. Characteristics o f abnormal waves i n north sea storm sea states. Appl Ocean Res 2003;25:337-44.

]21] Guedes Soares C, Cherneva Z, Antao EM. A b n o r m a l waves d u r i n g hurricane camille. J Geophys Res 2004; 109:1 - 7 . doi: 10.1029/2003JC002244. [C08008]. [ 2 2 | Guedes Soares C, Cherneva Z, Antao E. Steepness and a s y m m e t r y o f t h e largest

waves i n s t o r m sea states. Ocean Eng 2004;31:1147-67.

[ 2 3 | Guedes Soares C, Cherneva Z. Spectrogram analysis o f the t i m e - f r e q u e n c y characteristics of ocean w i n d waves. Ocean Eng 2005;32:1643-63.

[ 2 4 | Hasselmann K. On the non-linear energy transfer in a g r a v i t y wave spectrum. Part I.J Fluid Mech 1962;12:481-500.

[25] Hasselmann K. On t h e non-linear energy transfer i n a g r a v i t y wave spectrum. Part II. J Fluid Mech 1 9 6 2 ; 1 5 : 2 7 3 - 8 1 .

[26| Hasselmann K. On the non-linear energy transfer i n a g r a v i t y wave spectrum Part 111. J Fluid Mech 1963;15:385-98.

[27] Hasselmann K, M u n k W , McDonald G. Bispectra o f ocean waves. I n : Rosen-blatt M, editor. Time series analysis. NY: W i l e y ; 1963. p. 1 2 5 - 3 9 .

] 2 8 | Haver S. Freak waves - a suggested d e f i n i t i o n and consequences f o r marine structures. In: Proceedings rogue waves conference, IFREMER Brest. 2004. [29] Haver S, Karunakaran D. Probabilistic description o f crest heights o f ocean

waves. I n : Proc. o f t h e 5 t h international w o r k s h o p o n wave hindcasting and forecasting. 1998.

[30] HIawatsch F. Boudreaux-Bartels G. Linearand quadratic t i m e - f r e q u e n c y signal representations. IEEE Signal Process Mag 1 9 9 2 ; ( A p r i l ) : 2 1 - 6 7 .

131] Huang N . Shen Z, Long S. W u M . Shih H. Zheng a et al. The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis. Proc R Soc 1998;454:903-95.

]321 Kharif C. Pelinovsky E. Physical mechanisms o f t h e rogue w a v e phenomenon. European J Mech B/Fluids 2003;22:603-34.

[33] KwonSH, Lee HS. Kim CH. Wavelet transform based coherence analysis of freak wave and its impact. Ocean Eng2005;32(13):1572-89.

220 Z. Cherneva. C. Guedes Soares/ AppUed Ocean Research 30 (2008) 215-220

[ 3 4 ] K u r k i n A, Pelinovsky E. Freak waves: Facts, theory and modeling. Nizhny Novgorod: Nizhegorod Techn Univ Printing House; 2004. p. 158 [ i n Russian]. [35] Liu P. Wavelet spectrum analysis and ocean w i n d waves. In: Foufoula-Georgiou, Kumar, editors. Wavelets in geophysics. NY: Acad. Press; 1994. p. 1 5 1 - 6 6 .

]36] Liu P. Is the w i n d wave frequency spectrum outdated? Ocean Eng 2000;27: 5 7 7 - 8 8 .

[37] Liu P, M o r i N. Wavelet spectrum of freak waves i n the ocean. I n : Proc of proc 2 7 t h i n t conf on coastal eng ICCEOO. ASCE; 2000. p. 1092-8.

[ 3 8 ] Longuet-Higgins M . The effect of non-linearity on statistical distributions in the t h e o r y of sea waves. J Fluid Mech 1963; 17:459-80.

[39] Longuet-Higgins M . On the skewness of the sea surface slopes. J Phys Oceanogr 1 9 8 2 ; 1 2 : 1 2 8 3 - 9 1 .

[40[ Massel S. Wavelet analysis for processing of ocean surface wave records. Ocean Eng 2 0 0 1 ; 2 8 : 9 5 7 - 8 7 .

[41 ] M a r t i n W , Frost R. W i g n e r - V i l l e spectral analysis of non-stationary processes, IEEE Trans Acoust Speech Signal Process 1986;ASSP-33:1461-70.

[42] McComas H, Briscoe M . Bispectra of internal waves. J Fluid Mech 1980;97: 2 0 5 - 1 3 .

[ 4 3 | M o r i N, Liu PC, Yasuda N. Analysis of freak w a v e measurements in the sea of Japan. Ocean Eng 2002;29:1399-414.

[ 4 4 | Nikias CL, Raghuveer M . Bispectrum estimation: A digital signal processing f r a m e w o r k . Proc IEEE 1 9 8 7 ; 7 5 : 8 6 9 - 9 1 .

[45] Peryn F Prost R. A unified d e f i n i t i o n for the discrete- time, discrete-frequency and discrete t i m e - f r e q u e n c y W i g n e r distributions. IEEE Trans Acoust Speech Signal Process 1986;34:858-67.

[46] Petrova P. Cherneva Z, Guedes Soares C. Distribution of crest heights i n sea states w i t h abnormal waves. Appl Ocean Res 2006;28:235-45.

[47[ Petrova P. Cherneva Z, Guedes Soares C. On the adequacy o f second-order models to predict abnormal waves. Ocean Eng 2 0 0 7 ; 3 4 : 9 5 6 - 6 1 .

[48] Rodriguez GR, Guedes Soares C. Pérez Mai tell E, Nieto Borge J. N o n - u n i f o r m i t y in the w i n d generated gravity waves phase distribution. In: Proc 2 6 t h int conf on coastal e n g 1998. p. 3 6 6 8 - 7 9 .

[49] Slunyaev A, Pelinovsky E, Guedes Soares C. Modeling freak waves f r o m the north sea. Appl Ocean Res 2 0 0 5 ; 2 7 : 1 2 - 2 2 .

[50] Tayfun M , Lo J. Envelope, phase and narrow-band models of sea waves. J W t r w a y Post Coast and Oc Eng ASCE 1989;115(5):594-613.

[51] Veltcheva A, Guedes Soares C. Identification of the components of wave spectra by Hilbert Huang t r a n s f o r m method. Appl Ocean Res 2 0 0 4 ; 2 6 : 1 - 1 2 . [52[ Veltcheva A, Guedes Soares C Analysis of abnormal wave records by

the Hilbert Huang transform method. J Atmos Ocean Technol 2007;24: 1678-1689.